75677 ÷ 5 = 15135 R2
Alright, let's work through the division of 75677 by 5 step by step.

Step 1:
5 goes into 7 1 times with a remainder of 2.
Write down 1 as next digit of of the result. 
Result so far: 1
Subtract 5 from 7 to get 2.
Bring next digit (5) of the dividend behind the 2 and repeat the process: 25 / 5

Step 2:
5 goes into 25 5 times with a remainder of 0.
Write down 5 as next digit of of the result. 
Result so far: 15
Subtract 25 from 25 to get 0.
Bring next digit (6) of the dividend behind the 0 and repeat the process: 6 / 5

Step 3:
5 goes into 6 1 times with a remainder of 1.
Write down 1 as next digit of of the result. 
Result so far: 151
Subtract 5 from 6 to get 1.
Bring next digit (7) of the dividend behind the 1 and repeat the process: 17 / 5

Step 4:
5 goes into 17 3 times with a remainder of 2.
Write down 3 as next digit of of the result. 
Result so far: 1513
Subtract 15 from 17 to get 2.
Bring next digit (7) of the dividend behind the 2 and repeat the process: 27 / 5

Step 5:
5 goes into 27 5 times with a remainder of 2.
Write down 5 as next digit of of the result. 
Result so far: 15135
Subtract 25 from 27 to get 2.

The final result is 15135 with a remainder of 2.